Two data classes for a convolutional neural network, can one have a LOT more images for training than the other?
Your question is very general so therefore, in this case, my answer will be too: 

The answer is "sometimes": it depends on the data.

There can be a lot more images in one class than the other, and you can still get reasonable results. It highly depends on how much data you have of the "feature class".

If this is the case, we say that the classes are heavily unbalanced, and you need to do "class balancing". You do not want to do overfitting on this one class, and preferably you want the feature-class to be the biggest.

Another approach for CNNs is to use "dropout". Well, for CNN's you can go a bit further: you can remove parts of the image to generate "new" images. This way you prevent overfitting of the "feature" class, whilst generating more data.

I suspect that training all possible false positives is impossible without overfitting the network somehow.

Hope it helps, and give you some google pointers :)



Just FYI: You basically, in the tech term, want to know whether it works to do binary CNN classification using a heavily imbalanced dataset.